
From IPM to BIF to e-Lens simulations - Recent advancements in simulation code development
Dominik Vilsmeier / GSI
3rd IPM Workshop
J-PARC, Tokai, Japan
September 18th 2018
Second IPM Workshop

3rd IPM Workshop, September 18th 2018, D. Vilsmeier


pip install virtual-ipm
Second IPM Workshop

3rd IPM Workshop, September 18th 2018, D. Vilsmeier

Second IPM Workshop

3rd IPM Workshop, September 18th 2018, D. Vilsmeier

Gas Jet
BIF
2 Beams
+ DC Beam (e-)
BIF Monitor

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
🠖 Beam Induced
Fluorescence
Monitor

Vacuum gauge
N2 fluorescent gas
Beam
Viewport
Lens, Image-Intensifier, Camera
N2 gas jet
1) Excitation of gas by interaction with the beam
2) Register the corresponding light emission
Selection of gas type

3rd IPM Workshop, September 18th 2018, D. Vilsmeier

UV lines
lines with life times below 10 ns
Yellow lines
lines with life times about 20 ns
Blue lines
lines with life times about 60 ns
F. Becker et al 2009 "Beam Induced Fluorescence Monitor & Imaging Spectrography of Different Working Gases" Proc. of DIPAC'09
\(Ar^+\) lines with about 15 ns (currently being investigated)
Simulating BIF

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Particle generation
Ground state
Excitation rate is proportional to the number of beam particles

Particle detection
Tracking is stopped stochastically based on decay probability
N2 Gas jet

3rd IPM Workshop, September 18th 2018, D. Vilsmeier

Tilted gas sheet in order to capture both transverse dimensions

Projection of beam profile in the gas curtain plane
🠖 elliptical shape
x
y
z
y
Velocities are modeled according to Boltzmann distribution + additional jet component
Electron lens (e-Lens)

3rd IPM Workshop, September 18th 2018, D. Vilsmeier


x
y
z
y
p+
e-

- Hollow e-lens configuration for removing the beam halo
- Additional solenoid field along beam axis; 4T field strength in the center
- In reality BIF measurement won't be at the center
R. Veness 2017 "Beam-Gas Curtain (BGC) profile monitor: Project Overview and Status" 7th HL-LHC Collaboration Meeting
Electric fields

3rd IPM Workshop, September 18th 2018, D. Vilsmeier


z-axis
z-axis
Electric fields

3rd IPM Workshop, September 18th 2018, D. Vilsmeier


Though the average charge density seen by ions is dominated by e-beam the p-beam contributes significant spikes

Not to scale
e-Beam
10 keV CW @ 5 A
p-Beam
7 TeV
@ 40 MHz
e-Beam
p-Beam
Trajectories

3rd IPM Workshop, September 18th 2018, D. Vilsmeier

Ion trajectories for various starting points
4T solenoid field along beam axis
Ion motion mainly driven by solenoid field an e-beam electric field
Trajectories

3rd IPM Workshop, September 18th 2018, D. Vilsmeier

Zoom on trajectory @ starting point x = 1.2 mm
bunch center passing
p-Beam electric field becomes effective for small scale trajectories
Motion is still driven by other em-fields
Trajectories

3rd IPM Workshop, September 18th 2018, D. Vilsmeier

-
Trajectories simulated for 420 ns (decay time 60 ns)
-
Ions at greater starting position feel stronger e-field
- Ions are attracted by e-field while performing circular motion due to solenoid field
Trajectories

3rd IPM Workshop, September 18th 2018, D. Vilsmeier


Ion generated by the proton bunches close to their center are only influenced by the proton beam fields
Solenoid field still confines the movement but the e-field kick is much smaller
e-Lens Results

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Results for Nitrogen






e-Lens Results

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Results for Neon






Three-dimensional analytic bunch electric fields

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Original solution: P. Strehl, M. Dolinska, R.W. Müller (2000)
Initial implementation: M. Herty, P. Forck (2004)
Further development: S. Udrea, P. Forck (since 2015)
Elliptical bunch shape

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Ellipsoid:
Spheroid:
Consider a > b (i.e. bunches are longer than wide)
Parabolic charge distribution

RMS matched:
Electric field expression

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
• Using confocal elliptical coordinates:
• Transform Poisson equation:
• With the charge density:
Electric field expression

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Absolute field value inside the spheroid:
where
where
🠖 Polynomial expression of \(\xi, \eta\)
🠖 Similar for outside the spheroid but contains logarithmic terms
Numerical (in-)stability

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
The given expression is analytically correct however its implementation can become numerically instable for very large "a/b" ratios (i.e. very long bunches)
🠖 Observed for example for LHC 6.5 TeV bunch configuration
Implementation:

For LHC case the "a/b" ratio is around

Computed with mpmath library (1000 dps fps precision)
comp. time
Similar problem for the longitudinal electric field
all stacked
Radial field
Reformulation (inside bunch)

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Electric field inside the bunch:
Polynomials of (r, z)
🠖 no transformation to elliptical coordinates
No "dangerous" a/b ratios
Concise formulation
🠖 faster computation
(+ saves a few hundred lines of code)
Radial field
Longitudinal field
Comparison / Results

3rd IPM Workshop, September 18th 2018, D. Vilsmeier



Longitudinal field escalates already at smaller "a/b" ratios
old
new
New formulation provides numerically stable fields with less effort
Reformulation (outside bunch)

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
The electric field outside the bunch can be similarly simplified (e.g. longitudinal field):
term indicates numerical (in)stability (compute via e.g. xlog1py from scipy.special)
Can be used as a "safeguard" (should be > 0)
Transformation to elliptical coordinates remains crucial but works for much larger "a/b" ratios (up to \(10^8\); further improvements are under investigation)
Realistic "a/b" ratios are \(\lessapprox10^{7}\)
(LHC @ 14 TeV has \(a/b \approx 10^{7}\))
Reformulation (outside bunch)

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
Radial electric field outside the bunch

Again the logarithmic term can be used as an instability "safeguard"
Only limited by accuracy of coordinate transformation (allowing for much larger "a/b" ratios which cover well beyond the realistic regime)
Summary

3rd IPM Workshop, September 18th 2018, D. Vilsmeier
- New computational models for simulating BIF (+ gas jet)
- Simulation of multiple beams simultaneously (e-Lens)
- Additional models for different bunch shapes and corresponding electric fields (e.g. Hollow, Generalized Gaussian, Q-Gaussian)
- Good progress on analytic electric field models (for parabolic ellipsoid shape)
- Complete reformulation of electric fields inside and outside the bunch
- Provides numerically stable results with less effort
- Concise formulation results in increased performance
Available at ...

3rd IPM Workshop, September 18th 2018, D. Vilsmeier


... the Python package index:
... GitLab (source code + issue tracker):
... GitLab pages (documentation):
From IPM to BIF to e-Lens simulations
By Dominik Vilsmeier
From IPM to BIF to e-Lens simulations
International workshop on non-invasive beam profile monitors for hadron machines and its related techniques: 3rd IPM workshop, Japan Proton Accelerator Research Complex (Tokai, Japan), September 18th 2018
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